You will find lots of Chinese stir-fry dishes with rice in Hong Kong. With meat, fish or tofu and vegetables. As a big cosmopolitan city you can probably find all sorts of cuisines but we mostly ate in small canteen-style cafes in Kowloon. They don't always speak English but sometimes the menus have pictures which helps!
If you are vegetarian or vegan, then you will be well catered for. Traditionally, Buddhists don't eat meat, and the Chinese have a tradition of making fake meat that goes back a long way. Made from soya, mushroom and seitan these can be so convincing you will find yourself double checking whether it is real! It doesn't help that the menus in these restaurants refer to them as chicken, pork, duck etc. These Buddhist restaurants also happen to be great for Low FODMAP diets because they generally avoid onion and garlic. Eggs are allowed though, so watch out for that.
More recently, vegan food has become quite trendy in Hong Kong and you will find plant-based restaurants selling western-style vegan burgers and fake meat products. They can be quite expensive though.
Watch out for...
For some reason (perhaps from the British influence?) condensed milk is popular. This can appear where you least expect it, such as in soup!
In Malaysia you will find lots of curries with rice, very rarely containing dairy but more likely to have coconut milk. There were also a lot of noodle soup dishes made with rice noodles. Beef rendang is a favourite, which is a slow cooked beef dish with coconut, lemongrass and chilli. It can be very spicy!
Most places had satay - marianaded meat on skewers with a peanut dipping sauce. In one of the hotels we stayed in I had a satay cooking lesson. It is definitely dairy-free although I was quite shocked at how much sugar went into the sauce! (That's why it tastes so good...)
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Nasi lemak is a common breakfast dish consisting of coconut rice with a spicy sauce, and various other garnishes including egg.
In Vietnam you will find lots of rice, rice-noodles and stiry-fry type dishes. Similar to Chinese cuisine but slightly more aromatic with the use of mint and other herbs. A popular breakfast dish is pho which is made with rice-noodles in beef or chicken broth. You normally add your own meat, beansprouts, chilli, mint and fish sauce to taste.
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Most of the noodles are made of rice, but sometimes they are egg-noodles.
In Thailand you will find lots of rice, coconut milk based curries and stiry-fry type dishes. It's a great place to visit if you avoid dairy and eggs. The food can be very spicy. If you are vegan then vegetable stir fries are very common.
Mango Sticky Rice is amazing dessert consisting of fresh mango with sticky coconut rice and a sweet coconut sauce. It has no dairy or egg in it and is very, very tasy!
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Sometimes stir fry dishes contain egg, or come with a fried egg on top.
The food in Cambodia is similar to Thailand but less spicy. There are stir-fries and coconut milk based curries, with fish sauce and lemongrass. Everything comes with rice or rice-noodles. The local Kampot pepper is used to flavour dishes rather than chilli.
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Amok is a steamed fish dish which contains egg.